M1 June 2018 Q2
2.
A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at 30\(^\circ\). The coefficient of friction between the particle and the plane is \(\dfrac{1}{4}\). The particle is held in equilibrium by a force of magnitude \(P\) newtons. The force makes an angle of 20\(^\circ\) with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of \(P\). (10)
| Scheme | Marks |
|---|---|
| (Parallel to plane): \(\ P\cos 50 + F = 2g\cos 60\) | M1 A2 |
| (Perp to plane): \(\ R - P\sin 50 = 2g\cos 30\) | M1 A2 |
| \(F = \dfrac{1}{4}R\) | B1 |
| Attempt to eliminate \(F\) and \(R\) to give an equation in \(P\) only | M1 |
| Solve for \(P\) | DM1 |
| \(P = 6.7\) (2 SF) or 6.66 (3SF) | A1 |
| (10 marks) |
Notes
First M1 for resolving parallel to the plane with usual rules.
\(2g\) term must be using 30\(^\circ\) or 60\(^\circ\) angle but allow sin/cos confusion.
First and second A1’s for a correct equation. A1A0 if one error.
Second M1 for resolving perpendicular to the plane with usual rules.
\(2g\) term must be using 30\(^\circ\) or 60\(^\circ\) angle but allow sin/cos confusion.
Third and fourth A1’s for a correct equation. A1A0 if one error.
B1 for \(F = \frac{1}{4}R\) seen or implied
Third M1, independent but must have two 3 (or 4) term equations, for attempt to eliminate \(F\) and \(R\) to give an equation in \(P\) only.
Fourth DM1, dependent on third M1, for solving for \(P\).
Fifth A1 for 6.7 or 6.66
Other possible equations:
| \((\rightarrow)\): \(R\cos 60 - F\cos 30 = P\cos 20\) M1 A2 | |
| \((\uparrow)\): \(R\cos 30 + F\cos 60 = P\cos 70 + 2g\) M1 A2 |
First M1 for resolving horizontally with usual rules.
\(R\) term must be using 30\(^\circ\) or 60\(^\circ\) angle and \(F\) term must be using 30\(^\circ\) or 60\(^\circ\) angle but allow sin/cos confusion.
First and second A1’s for a correct equation. A1A0 if one error.
Second M1 for resolving vertically with usual rules.
\(R\) term must be using 30\(^\circ\) or 60\(^\circ\) angle and \(F\) term must be using 30\(^\circ\) or 60\(^\circ\) angle but allow sin/cos confusion.
Third and fourth A1’s for a correct equation. A1A0 if one error.